Lines in one plane that do not intersect are called parallel; we write a ∥ b. Segments and rays lying on parallel lines are also called parallel. Theorem: two lines perpendicular to the same line are parallel. So through a point O not on line a we can draw a parallel: drop the perpendicular OA to a, then draw b through O perpendicular to OA. Parallel axiom: in a plane, through a point not on a given line exactly one line parallel to it can be drawn (it is related to Euclid’s fifth postulate). Corollary: two lines parallel to a third are parallel to each other; proof by contradiction: if they met, two parallels to the third line would pass through that point.
Worked examples
The horizontal lines of a notebook are parallel; railway rails are also an example of parallel lines.
If a ∥ c and b ∥ c then a ∥ b. If they met, two parallels to c would pass through the meeting point — contradicting the axiom.
Class activity
“Find the parallels”: in the classroom find pairs of parallel segments (desk edges, window frames, board lines) and note them in your notebook.
Practice
1
How many lines parallel to a given line can pass through a point not on it?
1
2
a ⊥ c and b ⊥ c. What can be said about a and b?
a ∥ b
3
a ∥ c, b ∥ c, d ∥ b. What follows for a and d?
All are parallel to each other, so a ∥ d.
4
Why does the parallel axiom say “only one”?
It is an axiom: that a parallel can be drawn is proved, but that it is the only one is accepted without proof.